Modeling Hyster esis in CLIPrec.ce.gatech.edu/rec2006/presentations/REC2006_Wittenberg.pdf · Constr aint Logic Prog r amming (CLP) Hybr id Systems Using CLP(F) fo r Hybr id Systems

Modeling Hysteresis in CLIPDavid K. Wittenberg and Timothy J. Hickey

Brandeis University

REC2006-talk.tex – Modeling Hysteresis in CLIP – David K. Wittenberg and Timothy J. Hickey – 20/2/2006 – 20:46 – p. 1/2

Outline

ToolsInterval ArithmeticConstraint Logic Programming (CLP)

Hybrid SystemsUsing CLP(F) for Hybrid SystemsModelingAnalysis

Examples


Interval Arithmetic

[Moore, 1966]Introduced rigor to calculations on floating point numbers

Closed under + - x / (if interval doesn’t contain 0)Standard inequalities are all true.Distributive law does not hold


Constraint Logic Programming

[Jaffar, Lassez 1987] Logic programming - each resultstates that there is a proof of the resultUse constraints rather than equalities.powerful, Fastest method for some problemsgood for scheduling - used by sports leaguesDoesn’t give answer - just limits places where solution canbe


CLP(I)

[Hickey 2000]CLP over RealsCareful Rounding in IEEE 754 Floating Point| ?- {Xˆ2=2,X>0}.X = 1.41421356237309... ? ;no| ?-


Multiple Solutions

| ?- {Xˆ2=2}.X = [-1.41421356237309536751922678377,

1.41421356237309536751922678377] ? ;no| ?-


Multiple Solutions

| ?- {Xˆ2=2}.X = [-1.41421356237309536751922678377,

1.41421356237309536751922678377] ? ;no| ?-

| ?- {Xˆ2=2},solve_clip(queue,[X],0.000001).X = 1.41421356237309... ? ;X = -1.41421356237309... ? ;(10 ms) no


CLP(F) – ODE solver

[Hickey 2000]Extension of CLP(I)Uses Taylor Expansions with Remainder


CLP(F) example

Q(F,A,E) !F " H([0, 1]), F ! = F, F ([0, 1]) # [$100, 100],

F (0) = 1, F (A) = 2, F (1) = E

| ?- type([F],function(0,1)),{[ ddt(F,1)=F, F in [-100,100],eval(F,0)=1,eval(F,A)=2, eval(F,1)=E ]}.

A = .6931471... E = 2.7182818...


Hybrid Systems

[Fahrland 1970]

Analog part - changes state according to ODEsDigital part - changes state in 0 time


CLP(F) for Hybrid Systems

RigorousODEs translate directly to CLP(F)Handles non-linear ODEsLogical Semantics


Two Tank System

K3

f1(t)

f2(t)


ODEs for Two Tank System

f !1 =

!k1 $ k2

%f1 $ f2 + k3 f2 > k3

k1 $ k2

"(f1) f2 & k3

f !2 =

!k2%

f1 $ f2 + k3 $ k4%

f2 f2 > k3

k2

"(f1) $ k4

"(f2) f2 & k3


Code for Two Tank System

f !1 = k1 $ k2

"f1 $ f2 + k3 f2 > k3

f !2 = k2

"f1 $ f2 + k3 $ k4

"f2 f2 > k3

twotank(case1,X10,X20,T0,X11,X21,T1,[K1,K2,K3,K4]:- decls([X1,X2],function(T0,T1)),{[ ddt(X1,1) = K1 - K2*psqrt(X1-X2+K3),

ddt(X2,1) = K2*psqrt(X1-X2+K3) - K4*psqrt(X2eval(X1,T0)=X10, eval(X1,T1)=X11,eval(X2,T0)=X20, eval(X2,T1)=X21,X1 in [E,1000], X2 in [K3,1000], E=0.000000]}.


Analysis of Hybrid System

If tank levels X0 for the upper tank and Y0 for the lower satisfy0.62 & X0 & 0.63 ' 0.558 & Y0 & 0.567at time 0, then for all times n · 0.1 the tank levels X and Y satisfy

0.61922 & X & 0.63083 ' 0.55674 & Y & 0.56815

{X0 = [0.62,0.63],Y0=[0.558,0.567]},ks(Ks), twotank(case1,X0,Y0,0.0,X,Y,0.1,Ks)solve_clip(fwchk,[X,Y],N).

N=0 X=[.61931, .63069] Y=[.55697, .56802]N=1 X=[.61964, .63035] Y=[.55758, .56741]N=2 X=[.61985, .63013] Y=[.55796, .56703]N=3 X=[.61995, .63004] Y=[.55812, .56687]N=4 X=[.62000, .62999] Y=[.55819, .56680]


More Elegant Proof of Safety Property

| ?- {X10 = [0.62,0.63],X20=[0.558,0.567]},ks(Ks),twotank(case1,X10,X20,0.0,X11,X21,0.1,Ks),

({X11<0.62} ; {X11>0.63};{X21<0.558}; {X21>0.567}).

(1330 ms) no| ?-


Four Tanks

V1

V2

V3

V4

D2(t)

D3(t)

D4(t)

H1

H3

D1(t)

H4

H2


Valve Behaviour

0

2e-05

4e-05

6e-05

8e-05

0.0001

0.00012

0.00014

0.00016

0.00018

0.0002

0 0.2 0.4 0.6 0.8 1

f(x)


Enclosing Poor behaviour

-0.01 0.100

0.1 valve coefficentenclosure of valve coefficent

valve

FRACcoefficent

Valve position P

0.05


Equations For Each Tank

D!j(t) = Fj"1(t) $ Fj(t)

Rj(t) = Cj · eEj ·(1"Pj(t))3

Fj =

!Rj(t)

"Dj(t) $ Dj+1(t) + Hj+1 Dj+1(t) > Hj $ Hj+1

Rj(t)"

(Dj) Dj+1(t) & Hj $ Hj+1


CLP(F) Program For Each Tank

middle_tank(above,F1,D2,F2,D3,H2,KC2,FRAC2,H3,_E):- {[F2=KC2*FRAC2*psqrt(D2-D3+H),

ddt(D2,1)=F1-F2, H=H2-H3, D3 in [H,1000] ]}

middle_tank(near,F1,D2,F2,D3,H2,KC2,FRAC2,H3,E):- decls([EF],function(_,_)),{[F2=KC2*FRAC2*psqrt(D2-EF), ddt(D2,1)=F1-F2,H=H2-H3, D3 in [-2*E+H,2*E+H], EF in [0,2*E] ]

middle_tank(below,F1,D2,F2,D3,H2,KC2,FRAC2,H3,E):- {[F2=KC2*FRAC2*psqrt(D2),

ddt(D2,1)=F1-F2, H=H2-H3, D3 in [E,H]]}


Helper functions

valve_coef(normal,FRAC,P,KE) :-{[FRAC=exp(KE*((1-P))**3), FRAC in [0,1],

P in [0.01,1] ]}

valve_state_change(opening,trans,_P_before,opening,normal,P_after) :-

{P_after=0.01}.


Zeno Behavior

r2

h1

h2

i

o1 o2

r1


Hysteresis

Prevent “Chatter”Delay in switching statesOften used in physical systemsHere used both to reflect physics, and to prevent Zenobehaviour


Contributions of this Work

Improved Rigor in Dealing with Hybrid SystemsImproved Ease of Describing Hybrid SystemsModeling of Hybrid Systems with Non-Linear ODEsModeling of Points where ODEs are Non-AnalyticModeling of Areas with “Messy” or Poorly UnderstoodPhysicsImproved Ease of Analysis


Conclusions

CLP(F) is a powerful toolSimple Semantics – Easier argument that code iscorrectConstraint Approach allows one to work “backward”


Documents

Modeling Hyster esis in CLIPrec.ce.gatech.edu/rec2006/presentations/REC2006_Wittenberg.pdf · Constr aint Logic Prog r amming (CLP) Hybr id Systems Using CLP(F) fo r Hybr id Systems